Related papers: Abelian Complexity in Minimal Subshifts

We study a new notion of cyclic avoidance of abelian powers. A finite word $w$ avoids abelian $N$-powers cyclically if for each abelian $N$-power of period $m$ occurring in the infinite word $w^\omega$, we have $m \geq |w|$. Let…

In this paper we introduce and study a new property of infinite words: An infinite word $x\in A^\mathbb{N}$, with values in a finite set $A$, is said to be $k$-self-shuffling $(k\geq 2)$ if $x$ admits factorizations: $x=\prod_{i=0}^\infty…

Given a finite alphabet $\Sigma$ and a right-infinite word $\bf w$ over $\Sigma$, we define the Lie complexity function $L_{\bf w}:\mathbb{N}\to \mathbb{N}$, whose value at $n$ is the number of conjugacy classes (under cyclic shift) of…

We bound the number of distinct minimal subsystems of a given transitive subshift of linear complexity, continuing work of Ormes and Pavlov [7]. We also bound the number of generic measures such a subshift can support based on its…

We solve a case of the Abelian Exponential-Algebraic Closedness Conjecture, a conjecture due to Bays and Kirby, building on work of Zilber, which predicts sufficient conditions for systems of equations involving algebraic operations and the…

We prove surjectivity of certain word maps on finite non-abelian simple groups. More precisely, we prove the following: if N is a product of two prime powers, then the word map sending (x,y) to the product of the Nth powers of x and y is…

Generalizing the notion of automatic complexity of individual strings due to Shallit and Wang, we define the automatic complexity $A(E)$ of an equivalence relation $E$ on a finite set $S$ of strings. We prove that the problem of determining…

Shallit and Wang studied deterministic automatic complexity of words. They showed that the automatic Hausdorff dimension $I(\mathbf t)$ of the infinite Thue word satisfies $1/3\le I(\mathbf t)\le 2/3$. We improve that result by showing that…

We revisit the topic of power-free morphisms, focusing on the properties of the class of complementary morphisms. Such morphisms are defined over a $2$-letter alphabet, and map the letters 0 and 1 to complementary words. We prove that every…

The Thue-Morse set is the set of those nonnegative integers whose binary expansions have an even number of $1$. We obtain an exact formula for the state complexity of the multiplication by a constant of the Thue-Morse set $\mathcal{T}$ with…

Defant and Kravitz introduced generalizations of West's stack-sorting map $s$ from permutations to finite words. This raises questions as to how such generalizations could be applied in the field of combinatorics on words. The…

Let $\b$ be a Borel subalgebra of a simple Lie algebra $\g$ and let $\Ab$ denote the set of all Abelian ideals of $\b$. We consider $\Ab$ as poset with respect to inclusion, the zero ideal being the unique minimal element of $\Ab$. It was…

Two words $u$ and $v$ are $k$-abelian equivalent if, for each word $x$ of length at most $k$, $x$ occurs equally many times as a factor in both $u$ and $v$. The notion of $k$-abelian equivalence is an intermediate notion between the abelian…

We consider questions related to the structure of infinite words (over an integer alphabet) with bounded additive complexity, i.e., words with the property that the number of distinct sums exhibited by factors of the same length is bounded…

We prove 3SUM-hardness (no strongly subquadratic-time algorithm, assuming the 3SUM conjecture) of several problems related to finding Abelian square and additive square factors in a string. In particular, we conclude conditional optimality…

The symbolic complexity of an infinite word $W$ is the function $p_W(l)$ counting the number of different subwords in $W$ of length $l$. In this paper our main purpose is to study the complexity for a class of topological dynamical systems,…

Partial words are sequences over a finite alphabet that may contain wildcard symbols, called holes, which match or are compatible with all letters; partial words without holes are said to be full words (or simply words). Given an infinite…

Let $A$ be a simple abelian variety over a number field $k$ such that $\operatorname{End}(A)$ is noncommutative. We show that $A$ splits modulo all but finitely many primes of $k$. We prove this by considering the subalgebras of…

Each Abelian subgroup of the fundamental group of a compact and locally simply connected $d$-dimensional length space with no conjugate points is isomorphic to $\mathbb{Z}^k$ for some $0 \leq k \leq d$. It follows from this and previously…

Bell and Shallit recently introduced the Lie complexity of an infinite word $s$ as the function counting for each length the number of conjugacy classes of words whose elements are all factors of $s$. They proved, using algebraic…